Creatures that are dueling each other are spiritually linked to each other in a state that's slightly out of phase with everyone else. This doesn't have any inherent mechanical effects in and of itself, but it implies a structure for how those creatures interact with each other and the world around them that may be interacted with by duel modifying effects or other abilities. The term "in a duel" is often used to refer to a creature that is dueling. This does not mean that the "duel" must have been started by Duel.
If a creature that is dueling becomes incapacitated or attacks a creature that they are not dueling, this creature is considered to have "lost the duel". This also ends the dueling condition for that creature. If a creature is ever the only one remaining in a given duel, the duel ends and they are considered to have "won the duel".